Learn Finnish: Meillä ei edes ole puutetta, paitsi tätä pientä hävikkiä. - We do not even have a shortage, except for this small wastage.

It is governed by the word paitsi (except). When paitsi introduces an exception as a preposition, the noun phrase that follows it must be in the partitive case. This also naturally parallels the earlier partitive noun puutetta.

Could I use paitsi että in this sentence instead of just paitsi?

No, because paitsi että (except that) is a subordinate connector used to introduce a full clause with its own verb. Since tätä pientä hävikkiä is just a noun phrase without a verb, you must use the preposition paitsi.

Why is edes (even) placed directly between ei and ole?

In negative Finnish sentences, emphasizing adverbs like edes (even), aina (always), or koskaan (ever) are typically sandwiched right between the negation verb (here, ei) and the main verb (ole).

What kind of wastage does the word hävikki usually refer to?

Hävikki typically refers to material loss, inventory shrinkage, or waste, such as unsold food going bad in a supermarket or materials lost during manufacturing. It is a noun derived from the verb hävitä (to disappear / to lose).

As a quick refresher, why is (shortage) also in the partitive case here?